Copyright by Dr.Zheyan Jin Chapter 2 Some Fundamental Principles and Equations Road Map Aerodynamics tools Review of vector relations Basic flow equations Useful concepts for the implementation of the basic flow equations Continuity equation Substantial derivative Momentum equation Streamline Vorticity Energy equation Circulation Stream function Velocity potential
Copyright by Dr. Zheyan Jin Road Map Aerodynamics tools Basic flow equations Review of vector relations Continuity equation Substantial derivative Momentum equation Energy equation Circulation Vorticity Streamline Chapter 2 Some Fundamental Principles and Equations Chapter 2 Some Fundamental Principles and Equations Useful concepts for the implementation of the basic flow equations Velocity potential Stream function
Copyright by Dr.Zheyan Jin Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations A vector V in the plane or in space is an arrow:it is determined by its length,denoted V and its direction. Some Vector Algebra A+B=C A-B=D A.B=A Blcose A×B=(ABsin0E=G B 6 B G Vector Vector Dot Cross Addition Subtraction Product Product
Copyright by Dr. Zheyan Jin A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Chapter 2 Some Fundamental Principles and Equations Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations Some Vector Algebra B A V AB A Bcos A C V A B C A G B B B A D A B D Vector Addition Vector Subtraction A B A Bsin e G θ Dot Product θ e Cross Product
Copyright by Dr.Zheyan Jin Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations Typical Orthogonal Coordinate Systems ---Cartesian coordinate system y P (x,y,Z) f=xi+yj+z求 A=Ai+Ayj+Ak
Copyright by Dr. Zheyan Jin Chapter 2 Some Fundamental Principles and Equations Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations Typical Orthogonal Coordinate Systems j Az Ax A A i A j A k x y z i r k r xi yj zk --- Cartesian coordinate system P (x,y,z) y x z j i A k Ay
Copyright by Dr.Zheyan Jin Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations Typical Orthogonal Coordinate Systems ---Cylindrical coordinate system A X P(r ,z) x=rcos0 r=x+y A=A,e,+Age。+A,e2 y=rsine or 0=arctan Z=2 =2
Copyright by Dr. Zheyan Jin Chapter 2 Some Fundamental Principles and Equations Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations Typical Orthogonal Coordinate Systems Az r e e A r r z z A A e A e A e r k z z y r x r sin cos --- Cylindrical coordinate system P(r,θ,z) y x z A Ar θ z e z e r e e z z x y r x y arctan 2 2 or
Copyright by Dr.Zheyan Jin Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations Typical Orthogonal Coordinate Systems ---Spherical coordinate system e Z P,，0 A=A,e+Aoeo+A e Φ r=Vx2+y2+22 x=rsinecoso 0-arccos==arccos- z y=rsinesin Or x2+y2+2 z=rcos0 X arccos- x2+2
Copyright by Dr. Zheyan Jin Chapter 2 Some Fundamental Principles and Equations Chapter 2 Some Fundamental Principles and Equations 2.1 Review of Vector Relations Typical Orthogonal Coordinate Systems r e e A A e A e A e r r r cos sin sin sin cos z r y r x r --- Spherical coordinate system P(r,Φ,θ) z y x θ e A Ar A A e r e e 2 2 2 2 2 2 2 2 arccos arccos arccos x y x x y z z r z r x y z or Φ